I'm attempting to find the streamlines for a fluid flow given by the fluid velocity $\mathbf{u} = \begin{bmatrix}0 \\ -z + \cos{\omega t} \\ y + \sin{\omega t} \end{bmatrix}$.
I have formulated the 2 differential equations:
$$\frac{dx}{0}=\frac{dy}{-z+\cos(\omega t)}=\frac{dz}{y+\sin(\omega t)}$$
From this I know that $x=c_1$, where $c_1$ is an arbitrary constant. But how do I solve the rest?
Am I correct in saying that the remainder of the equation would yield:
$$(y+\sin{\omega t}) \space dy=(-z+\cos\omega t)\space dz$$
But this surely can't be right because I cannot integrate any $f(t)$ with respect to $y$ or $z$, can I?